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For example, in something like the General Utility Lattice Program, you perform structural optimisation and then use phonons to calculate properties like elastic constants etc.

The latter part seems to me to be quit obviously Lattice Dynamics, but would structural optimisation be Lattice Statics or Dynamics?

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Lattice statics refer more to the minimum energy lattice structure such as spacegroup, lattice constant, bond distances, or any other property you can derive from a geometric standpoint on that optimized structure. I don't think I would consider the actual optimization to be lattice statics, more of a way to get to the lattice statics.

Lattice dynamics such as phonon calculations help look at the excited states of the lattice to determine things such as thermal conductivity, infrared spectra, Raman spectra, or thermodynamic properties that depend on temperature.

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